{"id":5267,"date":"2022-08-19T01:22:31","date_gmt":"2022-08-19T01:22:31","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/?post_type=chapter&p=5267"},"modified":"2022-08-19T13:52:10","modified_gmt":"2022-08-19T13:52:10","slug":"10c-coreq","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/lumen-danacenter-statsmockup\/chapter\/10c-coreq\/","title":{"raw":"10C Coreq","rendered":"10C Coreq"},"content":{"raw":"In the next preview assignment and in the next class, you will need to be comfortable working with fractions, proportions, and percentages.\r\nParts of the Whole\r\nIn many types of research, we work with data that can be expressed with simple \u201cyes\u201d or \u201cno\u201d answers. Do you support the new law proposed in Congress? Does that patient have diabetes? Do cars come equipped with automatic braking? When we report the number of \u201cyes\u201d responses as a portion of the population or sample, we use a proportion.\r\n
\r\n

Question 1<\/h3>\r\nA small dairy farm is trying to decide whether or not to expand their operations. To\u00a0 gauge demand, they ask 100 different people a single question: \u201cDo you buy more\u00a0 than one gallon of milk per week?\u201d 20 people answer \u201cyes.\u201d\r\n
    \r\n \t
  1. One way to write this information would be as a fraction: [latex] \\frac{successes}{attempts} [\/latex]. Write the result from the survey in this format.<\/li>\r\n \t
  2. Convert your fraction to a decimal. We usually call this value the sample proportion, or [latex] \\hat{p} [\/latex].<\/li>\r\n \t
  3. Now, convert your decimal to a percentage.<\/li>\r\n<\/ol>\r\n<\/div>\r\n
    \r\n

    Question 2<\/h3>\r\nWhen working with proportions in research, the results may be written in any of these three formats. Complete the following table by converting between fractions, decimals, and percentages.\r\n\r\n<\/div>\r\n
    \r\n

    Question 3<\/h3>\r\nIn research, when using proportions, we can never have a result higher than [latex] \\hat{p} = 1 [\/latex].\u00a0 Consider our dairy farm example. If 100 people are surveyed, there is no way we\u00a0 can get more than 100 \u201cyes\u201d answers. This allows us to determine the proportion of\u00a0 failures for any given value of [latex] \\hat{p} [\/latex] (the proportion of successes in the sample) using the formula [latex] 1 - \\hat{p} [\/latex].\r\n
      \r\n \t
    1. Our sample proportion of \u201cyes\u201d answers in the dairy farm example is 0.2.\u00a0 What is the proportion of \u201cno\u201d answers?<\/li>\r\n \t
    2. The values of [latex] \\hat{p} [\/latex] and [latex] 1 - \\hat{p} [\/latex] are important parts of the formula we use to\u00a0 determine how much error there might be in our sample. Complete the\u00a0 following table to practice finding [latex] \\hat{p} [\/latex] and [latex] \\hat{p} [\/latex].\r\n
      \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      [latex] \\hat{p} [\/latex]<\/td>\r\n[latex] 1 - \\hat{p} [\/latex]<\/td>\r\n<\/tr>\r\n
      0.1<\/td>\r\n0.9<\/td>\r\n<\/tr>\r\n
      0.23<\/td>\r\n<\/td>\r\n<\/tr>\r\n
      <\/td>\r\n0.5<\/td>\r\n<\/tr>\r\n
      0.78<\/td>\r\n<\/td>\r\n<\/tr>\r\n
      0.91<\/td>\r\n<\/td>\r\n<\/tr>\r\n
      <\/td>\r\n0.85<\/td>\r\n<\/tr>\r\n
      <\/td>\r\n0.48<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div><\/li>\r\n<\/ol>\r\n<\/div>\r\n
      \r\n

      Question 4<\/h3>\r\nIt is also important to understand how changing the numerator or denominator of a\u00a0 fraction or proportion changes the overall value. Consider several different possible scenarios for the dairy farm example.\r\n
        \r\n \t
      1. Complete the following table by finding the missing values. The first row has been completed for you.\r\n
        \r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
        Number\r\n\r\nSurveyed<\/td>\r\nNumber of\r\n\r\n\u201cYes\u201d\r\n\r\nResponses<\/td>\r\nFraction<\/td>\r\n[latex] \\hat{p} [\/latex]<\/td>\r\n<\/tr>\r\n
        100<\/td>\r\n20<\/td>\r\n20\/100<\/td>\r\n0.2<\/td>\r\n<\/tr>\r\n
        100<\/td>\r\n40<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
        100<\/td>\r\n80<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
        50<\/td>\r\n20<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
        500<\/td>\r\n20<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
        1000<\/td>\r\n20<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div><\/li>\r\n \t
      2. As the numerator increases and the denominator is held constant, what\u00a0 happens to the value of the fraction?<\/li>\r\n \t
      3. As the denominator increases and the numerator is held constant, what\u00a0 happens to the value of the fraction?<\/li>\r\n \t
      4. Consider the fraction [latex] \\frac{x}{y} [\/latex]. As\u00a0[latex] x [\/latex] increases and [latex] y [\/latex] is held constant, what happens to the overall value of the fraction?<\/li>\r\n<\/ol>\r\n<\/div>\r\n ","rendered":"

        In the next preview assignment and in the next class, you will need to be comfortable working with fractions, proportions, and percentages.
        \nParts of the Whole
        \nIn many types of research, we work with data that can be expressed with simple \u201cyes\u201d or \u201cno\u201d answers. Do you support the new law proposed in Congress? Does that patient have diabetes? Do cars come equipped with automatic braking? When we report the number of \u201cyes\u201d responses as a portion of the population or sample, we use a proportion.<\/p>\n

        \n

        Question 1<\/h3>\n

        A small dairy farm is trying to decide whether or not to expand their operations. To\u00a0 gauge demand, they ask 100 different people a single question: \u201cDo you buy more\u00a0 than one gallon of milk per week?\u201d 20 people answer \u201cyes.\u201d<\/p>\n

          \n
        1. One way to write this information would be as a fraction: [latex]\\frac{successes}{attempts}[\/latex]. Write the result from the survey in this format.<\/li>\n
        2. Convert your fraction to a decimal. We usually call this value the sample proportion, or [latex]\\hat{p}[\/latex].<\/li>\n
        3. Now, convert your decimal to a percentage.<\/li>\n<\/ol>\n<\/div>\n
          \n

          Question 2<\/h3>\n

          When working with proportions in research, the results may be written in any of these three formats. Complete the following table by converting between fractions, decimals, and percentages.<\/p>\n<\/div>\n

          \n

          Question 3<\/h3>\n

          In research, when using proportions, we can never have a result higher than [latex]\\hat{p} = 1[\/latex].\u00a0 Consider our dairy farm example. If 100 people are surveyed, there is no way we\u00a0 can get more than 100 \u201cyes\u201d answers. This allows us to determine the proportion of\u00a0 failures for any given value of [latex]\\hat{p}[\/latex] (the proportion of successes in the sample) using the formula [latex]1 - \\hat{p}[\/latex].<\/p>\n

            \n
          1. Our sample proportion of \u201cyes\u201d answers in the dairy farm example is 0.2.\u00a0 What is the proportion of \u201cno\u201d answers?<\/li>\n
          2. The values of [latex]\\hat{p}[\/latex] and [latex]1 - \\hat{p}[\/latex] are important parts of the formula we use to\u00a0 determine how much error there might be in our sample. Complete the\u00a0 following table to practice finding [latex]\\hat{p}[\/latex] and [latex]\\hat{p}[\/latex].\n
            \n\n\n\n\n\n\n\n\n\n\n
            [latex]\\hat{p}[\/latex]<\/td>\n[latex]1 - \\hat{p}[\/latex]<\/td>\n<\/tr>\n
            0.1<\/td>\n0.9<\/td>\n<\/tr>\n
            0.23<\/td>\n<\/td>\n<\/tr>\n
            <\/td>\n0.5<\/td>\n<\/tr>\n
            0.78<\/td>\n<\/td>\n<\/tr>\n
            0.91<\/td>\n<\/td>\n<\/tr>\n
            <\/td>\n0.85<\/td>\n<\/tr>\n
            <\/td>\n0.48<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/li>\n<\/ol>\n<\/div>\n
            \n

            Question 4<\/h3>\n

            It is also important to understand how changing the numerator or denominator of a\u00a0 fraction or proportion changes the overall value. Consider several different possible scenarios for the dairy farm example.<\/p>\n

              \n
            1. Complete the following table by finding the missing values. The first row has been completed for you.\n
              \n\n\n\n\n\n\n\n\n\n
              Number<\/p>\n

              Surveyed<\/td>\n

              		Number of<\/p>\n

              \u201cYes\u201d<\/p>\n

              Responses<\/td>\n

              		Fraction<\/td>\n[latex]\\hat{p}[\/latex]<\/td>\n<\/tr>\n
              100<\/td>\n20<\/td>\n20\/100<\/td>\n0.2<\/td>\n<\/tr>\n
              100<\/td>\n40<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
              100<\/td>\n80<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
              50<\/td>\n20<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
              500<\/td>\n20<\/td>\n<\/td>\n<\/td>\n<\/tr>\n
              1000<\/td>\n20<\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/li>\n
            2. As the numerator increases and the denominator is held constant, what\u00a0 happens to the value of the fraction?<\/li>\n
            3. As the denominator increases and the numerator is held constant, what\u00a0 happens to the value of the fraction?<\/li>\n
            4. Consider the fraction [latex]\\frac{x}{y}[\/latex]. As\u00a0[latex]x[\/latex] increases and [latex]y[\/latex] is held constant, what happens to the overall value of the fraction?<\/li>\n<\/ol>\n<\/div>\n

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